The Kiss Precise

More on Descartes and his Circle Theorem

And on Frederick Soddy and his poetry

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For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. Four circles to the kissing come The smaller are the benter. The bend is just the inverse of The distance from the center. Though their intrigue left Euclid dumb, There's now no need for rule of thumb. Since zero bend's a dead straight line, And concave bends have minus sign, The sum of the squares of all four bends Is half the square of their sum.     To spy out spherical affairs An oscular surveyor Might find the task laborious, The sphere is much the gayer, And now besides the pair of pairs A fifth sphere in the kissing shares. Yet, signs and zero as before, For each to kiss the other four, The square of the sum of all five bends Is thrice the sum of their squares. Frederick Soddy (1936)     The Complex Kiss Precise Yet more is true: if all four discs Are sited in the complex plane, Then centers over radii Obey the self-same rule again. Wilks, Mallows and Laguna (2001)

Drag the centers to rearrange the circles- or use the table below to control location and curvature/radius. Try some Apollonian Gaskets of your own - and try the Quiz! JIGSAW Reverse QUIZ Gaskets EDIT MODE ⇓ ♬ RESET This document requires an HTML5-compliant browser. Gasket ▶ 0 0 50 Depth ▶ 0 0 100 Bend Radius Center(x) Center(y) App generated by GXWeb Show/Hide Fill Axis Labels

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Enter your name:	? Abel, Nils Agnesi, Maria Bernoulli, Daniel Cantor, Georg Descartes, Rene Euler, Leonhard Fermat, Pierre Galois, Evariste Germain, Sophie Hypatia Johnson, Katherine Liebniz, Gottfried Lovelace, Ada Monge, Gaspard Newton, Isaac Noether, Emmy Pythagoras Ramanujan, Srinivasa Riemann, Bernhard Sylvester, James Tao, Terence Weierstrass, Karl' Zeno
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